Let Zn denote the ring of integers modulo n. A permutation of Zn is a sequence of n distinct elements of Zn. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of Zn, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutatio...

The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in vie...

The intuitionistic linguistic set (ILS) is an extension of linguisitc variable. To overcome the drawback of using single real number to represent membership degree and non-membership degree for ILS, the concept of interval-valued intuitionistic linguistic set (IVILS) is introduced through representing the membership degree and non-membership degree with intervals for ILS in this paper. The oper...

In most arithmetic systems the speed is limited by the nature of the building block which makes logic decisions and by the extent to which decisions of low order numeric significance can affect results of higher significance. In this contribution, an improved DNA representation [A. Fujiwara, K. Matsumoto, W. Chen, Procedures for logic and arithmetic operations with DNA molecules, Int. J. Found....

The vague sum (product) of a finite number of real numbers and the vague integral multiples (powers) of real numbers based on a perfect vague addition (multiplication) operation are introduced, and some of their fundamental properties are pointed out. Furthermore, some examples are designed to show how they can be practically implemented. Keyword: Fuzzy arithmetic, Vague arithmetic, Vague arith...